An algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities with addition–min composition
Introduction
In this paper, we are interested in studying the fuzzy relation inequalities with addition–min composition: and the optimization system with minimizing a linear objective function subject to system (1): where , , , , and the operations ‘⋅’ and ‘+’ represent, respectively, the ordinary multiplication and addition, .
The notion of fuzzy relation equations with the max–min composition was first investigated by Sanchez [16], then various applications and the optimization models had been studied in [3], [4], [5], [6], [13], [18], [19], [22]. However, several studies [2], [14], [17], [20], [23] had shown that the max–min operator may not always be the most desirable fuzzy relational composition and in fact the max-product operator was superior in these instances. Since then, the theoretical results and applications also have been studied in [1], [7], [8], [10], [11], [12], [15].
System (1) was explored by Shao-Jun Yang and Jian-Xin Li [9]. The data transmission mechanism in BitTorrent-like Peer-to-Peer file sharing systems is reduce to system (1) in that paper. Suppose there are n users who are downloading some file data simultaneously in a BitTorrent-like P2P file sharing system. Let us label those . We investigate the conditions of the ith user receiving the file data from the other users. Suppose that the jth user sends the file data with quality level to , and the bandwidth between and is . Because of the bandwidth limitation, the network traffic that receives from is actually . Suppose that the quality requirement of download traffic of is at least , then we get the conditions of the ith user receiving the file data from the other users as follows: On the basis of (⁎), we add the in ith inequality, and normalize those variables. Let , then we get (1).
Although P2P is a rapid and efficient mode of transmission, it may cause network congestion when we use it to conduct a large-scale data transmission, such as a live broadcast of the Olympic tournament. In order to avoid network congestion and ensure data transmission, one of the optimal management objects quested by network operators is where is the quality level of the network traffic that sends to the other users. The general form of this equality is where , then we get the optimization system (2).
The aim of this paper is to find the optimal solution of system (2). In the paper, Sections 2 Preliminaries, 3 The minimal solutions of, 4 An algorithm about the pseudo-minimal indexes discuss the definitions and theorems that are prepared for our algorithm. Section 3 defines the pseudo-minimal indexes of system (1), and discusses properties of the minimal solutions. Section 4 gives an algorithm about the pseudo-minimal indexes. Section 5 obtains an algorithm for system (2).
Section snippets
Preliminaries
Let and be two index sets, then (1) can be tersely described as follows or where, , and .
Definition 1 (See [9].) Denote . Let , , we define: if , ; if and there are some such that .
In what follows we shall denote the dual of order relation “<” and “≤” by the symbol “>” and “≥”,
The minimal solutions of (1)
Hereinafter, we always assume that (1) is solvable, and in this part, we study some properties about the minimal solutions of (1).
For arbitrary , denote , and .
Theorem 4 System (1) has the unique minimal solution if and only if is a solution of (1), i.e. . In particular, when (1) has the unique minimal solution, is the unique minimal solution of system (1).
Proof Let . It follows from Theorem 2 that
An algorithm about the pseudo-minimal indexes
From Section 3, we can immediately see that the pseudo-minimal indexes play an important role in solving the minimal solutions when (1) has more than one minimal solution, so in this section, we present an algorithm to search the pseudo-minimal indexes, which is named PMI algorithm. should be computed before the algorithm. Denote at the begin of the algorithm.
PMI algorithm:
step.1 Choose arbitrary , if is not a solution of system (1), go to step.2, else if
The optimal solution of (2)
The optimal system (2) can be described as:
Lemma 7 If is the optimal solution of (2), then x is a minimal solution of (1), i.e. .
Proof This proof is trivial. □
According to Lemma 7, the optimal solution is a minimal solution of (1), and according to Theorem 5, there exist some pseudo-minimal indexes and minimal intervals s.t. , so it is sufficient to search the optimal solutions of system (2) in all minimal
Conclusion
The principal results of this paper is that we provide an algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities with addition–min composition. In order to obtain this algorithm, the nature of the minimal solution about the inequalities are analyzed, and also we define a new order in the solution space. Then we give the definition of the pseudo-minimal indexes and minimal intervals of this system, and through embedding system (2) into the minimal
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